1

Lecture 4

Visualization and Animation

2

Use Mathematica to visualize

the motion of physical

systems

Simple examples

Animate the following:

A freely moving particle bounded to a finite

line: 1D motion

A freely moving particle bounded to a square

box: 2D motion

A freely moving particle bounded to a square

rectangular box: 3D motion

𝑁 freely moving particle bounded to a square

rectangular box: 3D motion

Parametric equations

Given a set of parametric equation describing the motion of a particle in space as a function of time,

𝑥 = 𝑓 𝑡 , 𝑦 = 𝑔 𝑡

one can easily visualize the motion using the command 𝐌𝐚𝐧𝐢𝐩𝐮𝐥𝐚𝐭𝐞

To this end you may have to also invoke a For

loop for generating the time-dependent

coordinate variables before visualizing them.

𝐏𝐚𝐫𝐚𝐦𝐞𝐭𝐫𝐢𝐜𝐏𝐥𝐨𝐭[] is another useful command

for this purpose.

4

Examples of parametric equations

SHO𝜃 𝑡 = 𝜃0 sin(𝜔𝑡 + 𝜙) ;

𝑥 = 𝐿 sin(𝜃(𝑡)) , 𝑦 = 𝐿 − 𝐿 cos(𝜃(𝑡))

O

+𝑥

𝜔 =𝑔

𝑙

+𝑦

Derivation of the equation of motion for SHO

Equation of motion (EoM)

Force on the pendulum

2; Tl

g

for small oscillation,

𝐹𝜃 = 𝑚𝑎𝜃

−𝑚𝑔 sin 𝜃=𝑚𝑑𝑣𝜃

𝑑𝑡= 𝑚

𝑑

𝑑𝑡

𝑑𝑟

𝑑𝑡≈ 𝑚

𝑑2

𝑑𝑡2𝑙𝜃

𝑑2𝜃

𝑑𝑡2≈ −

𝑔𝜃

𝑙= −𝜔2𝜃

𝜃 𝑡 = 𝜃0 sin(𝜔𝑡 + 𝜙)

r

l

Animation of simple pendulum

7

Examples of parametric equations:

2D projectile motion The trajectory of a 2D projectile with initial

location (𝑥0, 𝑦0), speed 𝑣0 and launching

angle 𝜃 are given by the equations:

𝑥 𝑡 = 𝑥0 + 𝑣0𝑡 cos 𝜃;

𝑦 𝑡 = 𝑦0 + 𝑣0𝑡 sin 𝜃 +𝑔

2𝑡2

for t from 0 till T, defined as the time of flight,

T =-2(𝑦0 + 𝑣0 sin 𝜃)/𝑔.

𝑔 = −9.81;

𝑥

𝑦+𝑥

+𝑦

O

Animation of 2D projectile

10

Consider a planet orbiting the Sun which islocated at one of the foci of the ellipse.

The coordinates of the planet at time t can be expressed in parametrised form:

𝑥(𝑡) = ℎ + 𝑎cos(𝜔0𝑡), 𝑦 𝑡 = 𝑘 + 𝑏sin 𝜔0𝑡

Examples of parametric equations:

2-body Planetary motion

C(h,k)

ba

Animation of planetary orbital motion

12

Assignments

By using the corresponding parametric

equations for the (𝑥, 𝑦) coordinates,

1. Animate a SHO (w/o drag force and driving

force)

2. Animate 2D projectile motion (w/o drag force

and driving force)

3. Animate 2-body planetary motion

1D sinusoidal wave A sinusoidal wave is fundamentally

characterized by two quantities: wave number (equivalent to wave length) and angular frequency (frequency)

𝑘𝑥, 𝜔 or 𝜆, 𝑓 .

Wave number 𝑘𝑥 =2𝜋

𝜆; 𝜆 wave length

Angular frequency 𝜔 = 2𝜋𝑓; 𝑓 frequency

14

1D sinusoidal wave 𝜓 𝑥, 𝑡 = 𝐴 sin 𝜃(𝑥, 𝑡) ;

𝜃 𝑥, 𝑡 = 𝑘𝑥𝑥 − 𝜔𝑡 + 𝜙

Phase velocity of the wave can be obtained by

imposing the condition:𝜕𝜃(𝑥, 𝑡)

𝜕𝑡= 0

⇒ 𝑘𝑥𝜕𝑥

𝜕𝑡− 𝜔 = 0

⇒ 𝑣𝑥 =𝜕𝑥

𝜕𝑡=

𝜔

𝑘𝑥= 𝑓𝜆

𝑣𝑥 =𝜔

𝑘𝑥implies the wave is moving to the +𝑥

direction. 15

1D sinusoidal wave

If 𝜓 𝑥, 𝑡 = 𝐴 sin 𝜃(𝑥, 𝑡) , with

𝜃 𝑥, 𝑡 = 𝑘𝑥𝑥 + 𝜔𝑡 + 𝜙,

⇒ 𝑣𝑥 = −𝜔

𝑘𝑥

This wave is moving to the -𝑥 direction.

16

1D sinusoidal wave moving in +𝑥direction

17

Adding two 1D sinusoidal waves

Same amplitude, same frequency; To ignore all phases 𝜙(by setting 𝜙 = 0).

Same / opposite directions

Same / different wavenumbers

Same / different angular frequencies

𝜓0 𝑥, 𝑡 = 𝐴 sin 𝜃0(𝑥, 𝑡) ; 𝜓1 𝑥 = 𝐴 sin 𝜃1(𝑥, 𝑡) ;

𝜃0 𝑥, 𝑡 = 𝑘𝑥,0𝑥 ± 𝜔0𝑡;

𝜃1 𝑥, 𝑡 = 𝑘𝑥,1𝑥 ± 𝜔1𝑡= (𝑘𝑥,0+Δ𝑘𝑥,1)𝑥 ± (𝜔0 + Δ𝜔1)𝑡

Δ𝑘𝑥,𝑖 = 𝑘𝑥,𝑖 − 𝑘𝑥,𝑖−1; 𝑘𝑥,𝑖 = 𝑘𝑥,0 + 𝑖Δ𝑘𝑥,𝑖

Δ𝜔𝑖 = 𝜔𝑖 −𝜔𝑖−1; 𝜔𝑖 = 𝜔0 + 𝑖Δ𝜔𝑖;

Usually, Δ𝜔𝑖= Δ𝜔, Δ𝑘𝑥,𝑖= Δ𝑘𝑥.18

Animation exercises1. Based on the simple equation of a 1D sinusoidal wave:

i. Animate two 1D waves, one in the +𝑥 and another in -𝑥. Display both on the same graph, without interfering each other.

ii. Repeat 1 with both in the same directioniii. Repeat 1 with both waves are added to interfereiv. Repeat 2 with both waves are added to interfere

Two sinusoidal waves, moving in the same direction,

without interference, with a difference in wavenumber

and angular frequency of 5Δ𝑘 and 5Δ𝜔; 𝑘0 = 1;𝜔0 = 1;Δ𝑘=Δ𝜔 = 1/50;

20

Superposition of two sinusoidal waves, moving in the

same direction, with a difference in wavenumber and

angular frequency of 5Δ𝑘 and 5Δ𝜔; 𝑘0 = 1;𝜔0 = 1;Δ𝑘=Δ𝜔 = 1/50;

21

Superpositioning 𝑁 1D sinusoidal waves

Simulate the motion of the resultant wave form obtained from the superposition of 𝑁 waves with the following conditions:

All waves have the same amplitude 𝐴 and moving in the same direction; Ignore all phases 𝜙 (set all 𝜙 = 0);

Each wave has a different wavenumber and angular frequency:

𝜓𝑖 𝑥 = 𝐴 sin 𝜃𝑖(𝑥, 𝑡) ;

𝜃𝑖 𝑥, 𝑡 = 𝑘𝑥,𝑖𝑥 ± 𝜔𝑡;

𝑘𝑥,𝑖 = 𝑘𝑥,0 + 𝑖Δ𝑘𝑥,𝑖; 𝜔𝑖 = 𝜔0 + 𝑖Δ𝜔𝑖;

Fix initial values: 𝐴 = 1,𝜔0 = 1, 𝑘𝑥,0 = 1, 𝑁 = 25, Δ𝑘𝑥 =1

75, Δ𝜔 =

1

75.

Simulate for a total duration of 500 𝑇 𝑇 =2𝜋

𝜔0; Width of the

simulation box set to [−100𝜆0 , +100𝜆0 ]

22

23

Animation exercises1. Animate an outgoing 2D sinusoidal wave

2. Animate two outgoing 2D sinusoidal waves from two different origins that display interference.

More complicated examples

Damped, forced SHO

2D projectile motion with drag force from the

air

Three-body planetary motion

25